Question
If a tree is cut down to a stump will it eventually regrow into a tree?

Answer
It depends on the species of the tree and the size. Hardwood trees for the most part will sprout from the roots assuming the roots are alive. Conifers do not sprout. Once a tree gets real large they sprout less than a younger tree. If the tree was dead when cut the root system is also dead. What you will get  is many sprouts from the same stump and over time one will usually grow larger than the other. Once you can determine which sprout will be the largest you can cu the other leaving a one stem tree. Sprouts from a living older stump will grow fairly fast since they have a well established root system.